If a person takes out the SIM card, does that automatically mean that you have purchase a new SIM card to replace the old one.

Generally no.

If you put the SIM back in it will normally function as-before, and it will also generally be functional if placed in another device as well.

Note however that carrier might restrict a SIMs general portability as they program the SIM.

Also some devices might also be “carrier locked” meaning that they will only accept SIMs from a particular carrier.

Taking out SIM cards can be very complicated so I highly recommend doing research on it before doing anything. You also might have to unlock your phone. And I don’t mean password unlock, it is a different type of unlock that you have to go through online. I do not know much about it. Do some research